Two-spinor tetrad and Lie derivatives of Einstein-Cartan-Dirac fields

TL;DR

This paper introduces a unified geometric framework for Lie derivatives of spinors, connections, and gravitational fields within Einstein-Cartan-Dirac theory, based on minimal geometric data from a complex vector field.

Contribution

It presents a novel geometric formulation of Einstein-Cartan-Dirac fields using minimal data, enabling well-defined Lie derivatives without special assumptions on vector fields.

Findings

Lie derivatives are well-defined for all objects without special assumptions.

The framework maintains natural mutual relations among derivatives.

The approach simplifies the geometric treatment of Einstein-Cartan-Dirac fields.

Abstract

An integrated approach to Lie derivatives of spinors, spinor connections and the gravitational field is presented, in the context of a previously proposed, partly original formulation of a theory of Einstein-Carta-Maxwell-Dirac fields based on "minimal geometric data": all the needed underlying structure is geometrically constructed from the unique assumption of a complex vector field $S\to M$ with 2-dimensional fibers. The Lie derivatives of objects of all considered types, with respect to a vector field $X:M\to TM$, are well-defined without making any special assumption about $X$, and fulfill natural mutual relations.